Ulam-Type Approximations of Transfer Operators

• Ulam-type approximations are methods that replace infinite-dimensional transfer operators with finite matrices, facilitating the numerical study of dynamical systems.
• They provide rigorous convergence and error estimates for invariant measures and spectral data, aiding analysis in chaotic, random, and high-dimensional systems.
• Extensions using polynomial, wavelet, and optimal transport strategies enhance accuracy and address challenges like boundary biases and high-dimensional complexity.

Ulam-type approximations of transfer operators are a foundational technique in the numerical analysis of dynamical systems. These methods replace infinite-dimensional transfer or Perron–Frobenius operators by finite-rank, typically matrix, approximations constructed from suitable partitions or function bases. This approach enables rigorous and computationally tractable exploration of ergodic properties, invariant measures, spectral data, and statistical characteristics of dynamical systems, both deterministic and random. The thread of Ulam-type discretizations runs through diverse domains, from statistical mechanics to fluid dynamics, and underpins many concrete computational algorithms at the heart of contemporary dynamical systems research.

1. Principles of Ulam-type Approximation

The classical Ulam method involves partitioning the phase space $X$ into finitely many measurable subsets (or bins) $B_1,\dots,B_N$. The transfer operator $\mathcal{P}$—which advances densities or measures under a map $T$—is projected onto the space spanned by the indicator functions of these bins. The resulting finite-rank operator, represented as an $N \times N$ matrix $P$, typically has entries

$P_{ij} = \frac{m(B_i \cap T^{-1}(B_j))}{m(B_i)},$

where $m$ denotes a reference measure (e.g., Lebesgue). The matrix approximation acts on vectors representing discretized densities as

$\vec{\rho}_{n+1} = P\,\vec{\rho}_n.$

This approach—commonly termed the Ulam-Galerkin method—interprets transfer operator approximation as histogram-based density estimation and belongs to the broader class of Galerkin or Petrov-Galerkin discretizations (Surasinghe et al., 2022).

Extensions of Ulam's method include approximations on function spaces beyond characteristic functions (e.g., polynomial bases, wavelets), with atomic decomposition and Lagrange interpolation-based schemes offering alternative projection strategies that can exploit higher degrees of regularity or analytic structure (Arbieto et al., 2019, Bandtlow et al., 2020).

2. Convergence and Error Analysis

Rigorous analysis shows that Ulam-type approximations converge to the true spectral and statistical data of the original operator under suitable dynamical and functional analytic assumptions. For piecewise expanding maps with bounded variation, convergence rates for invariant densities are typically $O(\log N / N)$ in the $L^1$ norm, hinging on the Lasota-Yorke inequality and spectral gap properties of the transfer operator (Bahsoun et al., 2014). When observables or densities are more regular (e.g., holomorphic), interpolation-based discrete operators can achieve exponential convergence of spectral data due to complex contraction properties (Bandtlow et al., 2020).

Recent work recasts Ulam's method within the framework of statistical density estimation, enabling a rigorous quantification of bias, variance, and mean square error. The histogram estimator inherent in classical Ulam's method yields $O(N^{-2/3})$ mean square error for optimally chosen bin numbers, while kernel density estimation (KDE) approaches can achieve $O(N^{-4/5})$ error, with KDE outperforming histograms except near boundaries or jump discontinuities (Surasinghe et al., 2022).

For systems with trajectories rather than explicit maps, triangulation or grid-based approximations of the transfer operator have been shown to provide robust estimates of invariant measures and derived statistical quantities—even for sparse or noisy data (Diego et al., 2018).

3. Spectral and Statistical Applications

Ulam-type discretizations capture key spectral properties of dynamical systems, including:

• Invariant Densities: Fixed points of the approximated matrix operator correspond to discretizations of the system's invariant measure. Rigorous error bounds can be derived (Bahsoun et al., 2014, Surasinghe et al., 2022).
• Diffusion Coefficients and Statistical Quantities: Using the matrix approximation, one can estimate quantities such as diffusion coefficients arising from central limit theorems, with explicit error control via truncation and discretization (Bahsoun et al., 2014).
• Spectral Properties: Peripheral and subleading eigenvalues of the finite-rank approximation reflect mixing rates, decay of correlations, and resonance phenomena. Self-adjoint and quasi-compact treatments enable matrix-based spectral computations, with rigorous connections to the continuous spectrum (Ammou et al., 2014, Ammou et al., 2015).
• Coherent Structures and Ergodic Partitions: Rolling window SVDs of matrix products extract finite-time coherent modes, and tracking Oseledets spaces via singular vectors reveals merging, splitting, and persistence of spatial structures in non-autonomous systems (Blachut et al., 2020).
• Dynamical Quantities from Data: Transfer operator approximations estimated directly from trajectory data (via triangulation, grid, or regularized optimal transport) allow for robust computation of transfer entropy and identification of almost-invariant sets in molecular and high-dimensional dynamics (Diego et al., 2018, Junge et al., 2022).

4. Extensions Beyond Standard Piecewise Constants

The development of polynomial and atomic-basis discretizations extends Ulam's philosophy by projecting transfer operators onto function spaces with richer structure:

• Polynomial Eigenfunctions: Transfer operators associated to self-similar or affine IFS admit polynomial eigenfunctions, enabling explicit computation of moments, measures, and spectral data (Bandt et al., 2015).
• Atomic Decomposition: Besov-type spaces built from atomic decompositions provide a framework for discretization and analysis of operators with low regularity, supporting exponential decay of correlations and the CLT under weak regularity assumptions (Arbieto et al., 2019).
• Lagrange and Chebyshev Approximations: Interpolatory projections onto global polynomial bases can achieve exponential spectral convergence when the dynamics admit analytic extension, outperforming standard Ulam methods for systems with high smoothness (Bandtlow et al., 2020).

5. Recent Developments: Regularization, Residuals, and Spectral Pollution

Recent research has introduced refined discretization and regularization schemes to address spectral pollution and improve stability:

• Entropic Regularization and Optimal Transport: Embedding discretization in the theory of entropically regularized optimal transport yields compact, Markovian operators with provable convergence of spectra, stability of eigenfunctions, and robust performance on high-dimensional and data-based problems (Junge et al., 2022).
• Residual-based Filtering and Spectral Pollution: Spectral pollution, wherein discretizations introduce spurious eigenvalues not reflecting the infinite-dimensional operator, is addressed via residual-based screening. Approaches such as ResDMD and Kernel-EDMD test whether candidate eigenpairs nearly satisfy the operator eigenvalue equation in norm, filtering out false eigenvalues and ensuring accurate spectral approximation (Herwig et al., 22 Jul 2025).
• Function Space Selection and Functional-Analytic Subtleties: The spectrum of transfer and Koopman operators depends heavily on the function space considered (e.g., $L^2$, Hardy, or Sobolev spaces). Recent work demonstrates that approximations must be tailored to the analytic context to avoid missing genuine eigenvalues or producing spurious ones, and that the choice of function space critically impacts which global dynamical features are captured (Herwig et al., 22 Jul 2025).

6. Connections to Markov Processes and Wavelets

Transfer operators are fundamentally linked to Markov chains, and Ulam-type discretizations may be viewed as constructing finite-state Markov models for the underlying dynamics. Harmonic functions (operator eigenfunctions with eigenvalue 1) correspond to invariant densities or measures, and spectral properties of discretized transfer operators reflect the essential Markovian structure (Alpay et al., 2016). This connection extends to applications in wavelet multiresolution analysis, where transfer operators underpin scaling relations and the multiscale decomposition of signals (Alpay et al., 2016).

7. Limitations and Outlook

While Ulam-type approximations have broad applicability, they present limitations:

• Curse of Dimensionality: Direct grid-based discretizations become rapidly infeasible in high dimensions, motivating development of adaptive, kernel-based, or sampling-driven approaches (e.g., entropic transport, triangulation) (Junge et al., 2022, Diego et al., 2018).
• Boundary and Regularity Issues: Histogram-based Ulam methods can incur increased bias at boundaries or near discontinuities; kernel-based or polynomial approximations may alleviate some issues at the expense of introducing trade-offs between smoothness and fidelity (Surasinghe et al., 2022).
• Intermittency and Nonuniform Hyperbolicity: For systems with weakly expanding or intermittent behavior, the convergence and spectral characteristics of Ulam approximations require careful treatment, and may necessitate more sophisticated functional frameworks (Ammou et al., 2015).

Continued research is focused on extending Ulam-type methods to more complex systems—non-autonomous, high-dimensional, randomly forced, or with low regularity—while ensuring rigorous convergence for both statistical and spectral properties. Techniques such as spectral regularization, function space adaptation, and residual-based validation represent a growing toolkit supporting robust and accurate numerical operator theory across the spectrum of contemporary applications.